What exactly is not the case? Most humans have been dependent on animals for their diet. Even with rich narrative traditions, they would still have to limit their connection to the food, otherwise they would not eat it (similar to some people today). It is abnormal to have easily available food in part due to specialization (butchers, etc) and industrialization.

Are people likely to be desensitized by not being part of the process of getting the animal to the table? Possibly, but that is an ad-don to the points I made.